class: title-slide background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo-white.svg"), url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/title-image1.jpg") background-position: 10% 90%, 100% 50% background-size: 160px, 50% 100% background-color: #0148A4 # .text-shadow[.white[STAT5003]] ## .white[Week 10 : Monte Carlo] ### .white[Dr. Justin Wishart] ### .white[Semester 2, 2020] --- ## Readings <i class="fas fa-university fa-pull-right "></i> * Easily accessible text in Shonkwiler and Mendivil (2009) --- class: segue-yellow background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo.svg") background-size: 20% background-position: 95% 95% # Monte Carlo Methods --- ## What are Monte Carlo methods? * Monte Carlo are a class of computational methods that can be applied to a wide range of problems * Key aspect of Monte Carlo methods are that they rely on __random sampling__ * Generally provide approximate solutions - not exact solutions * Used in cases where analytical solutions don’t exist or are too difficult to implement * Monte Carlo methods are sometimes referred to as stochastic simulation --- ## History of the Monte Carlo method .columns-2[ .pull-left[ * Invented by a Polish mathematician called Stanislaw Ulam in the late 1940s * Inspiration during recovery from illness with a thought experiment: .scriptsize[ > The question was what are the chances that a Canfield Solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. ] * Monte Carlo was the code name given for the project. - Inspired by the Monte Carlo casino in Monaco where Ulam's uncle would borrow family money to gamble. ] .pull-right[ <img src="https://upload.wikimedia.org/wikipedia/commons/8/8a/STAN_ULAM_HOLDING_THE_FERMIAC.jpg" width="80%"/> ] ] --- ## Monte Carlo estimation of `\(\pi\)` .columns-2[ .pull-left[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> ] .pull-right[ * Area of circle `\(= \pi r^2\)` and area of square `\(= 4r^2\)` - `\((r\)` is the radius) * A Monte Carlo method to estimate `\(\pi\)`: 1. Randomly draw `\(N\)` points within unit square at random 2. Count the `\(R\)` points which are inside the unit circle 3. Compute the ratio `\(R/N\)` and estimate `\(\pi\)` as `\(4R/N\)` ]] --- ## Using Monte Carlo to estimate `\(\pi\)` In this example, the mathematical statistics of the procedure is: * Write the parameter we want to estimate `\((\pi\)` here) as an expectation * Represent the parameter as a sample approximation `\begin{align*} \mathbb{E} X = \int t f(t) \, dt \approx \frac{1}{N} \sum_{i = 1}^N X_i \end{align*}` * The law of large numbers .brand-red[guarantees] that as the number of samples we draw increase, the parameter converges to the true parameter value. --- ## Expectation of a random variable From last slide, we can write the expected value of a random variable `\(X\)` as a sum: `\begin{align*} \mathbb{E} X = \int_A t\cdot f(t)\, dt \approx \frac{1}{N} \sum_{i = 1}^N X_i \end{align*}` We can replace `\(X\)` with a function `\(g(X)\)`. Then the previous equation becomes: `\begin{align*} \mathbb{E} g(X) = \int_A g(t)\cdot f(t)\, dt \approx \frac{1}{N} \sum_{i = 1}^N g(X_i) \end{align*}` Where we assume to draw `\(x_i \sim f\)` --- ## Monte Carlo Integration example Goal is to evaluate the following definite integral `\begin{align*} \int_0^1 e^{-x^2/2}\, dx (\approx \frac{1}{N} \sum_{i = 1}^N g(X_i)). \end{align*}` Find a random variable such that the above holds. Need to be careful such that, * Domain of the random variable and `\(g\)` agree with the above definition. * One example is, - `\(g(x) = e^{-x^2/2}\)` - `\(X_i\)` is a uniform random variable on (0,1) - Then `\(\mathbb{E}g(X) = \int_0^1 e^{-x^2/2}\, dx \approx \frac{1}{N}\sum_{i = 1}^n g(X_i)\)` --- ## Monte Carlo visually <img src="https://github.com/usydstat5003/stat5003-graphics/blob/master/Week10/possible-visual-MC.png?raw=true" width="80%"/> --- ## Simulating random variable * Let’s say we know how to simulate random variables that has a uniform distribution on some interval `\(U \sim [0,1]\)` - Can do that in `R` with `runif` > How do we simulate random variables that can have any probability distribution? In `R`, we can draw random Gaussian variables using the function `rnorm` but how do these functions really work? Two methods (others exist): 1. Inverse-transform method 2. Acceptance-rejection method --- ## Inverse transform method * Let’s say `\(X\)` is a random variable that has a cumulative distribution function (cdf) `\(F\)`. * Remember that cdf is the integral of the pdf i.e. `\begin{align*} F(x) = \int_{-\infty}^x f(t) \, dt \end{align*}` * If the inverse of `\(F(x)\)` exists, then we can generate `\(X\)` as: `\begin{align*} X = F^{-1}(U) \end{align*}` --- ## Example - Exponential distribution The exponential distribution has a pdf of the form: `\begin{align*} f(x) = \begin{cases}\lambda e^{-\lambda x} & \quad \text{if } x > 0\\ 0 & \quad \text{if }x \le 0\end{cases} \end{align*}` and a cdf of the form: `\begin{align*} F(x) = \begin{cases}1 - e^{-\lambda x} & \quad \text{if } x > 0\\ 0 & \quad \text{if }x \le 0\end{cases} \end{align*}` The inverse cdf is: `\begin{align*} F^{-1}(x) = -\frac{\log(1 - x)}{\lambda} \end{align*}` To sample from the exponential distribution, we can do the following: 1. Sample `\(U \sim \text{Uniform}(0,1)\)` 2. Set `\(X = −\frac{\log(1 - U)}{\lambda}\)` --- ## Acceptance-Rejection method * The problem with the inverse transform method is that you need to know the functional from of the inverse of the cdf * Acceptance-rejection method is a more general method can simulate 'difficult' distributions --- ## Acceptance rejection method * Given two probability densities `\(f(x)\)` and `\(g(x)\)`, with `\(f(x) < Mg(x)\)` for all `\(x\)` * If `\(g(x)\)` is a distribution that can be easily sampled, then we can sample `\(f(x)\)` using the following procedure: 1. Draw a random variable `\(X \sim g(x)\)` 2. Accept `\(X\)` with probability `\(\displaystyle \frac{f(x)}{Mg(x)}\)` 3. Repeat steps 1 and 2 until you have the desired number of random samples --- ## Acceptance-rejection plot <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Problems with the Acceptance-rejection method * The envelop distribution `\(g(x)\)` needs to closely resemble `\(f(x)\)` for this method to work well * If `\(g(x)\)` does not match `\(f(x)\)` well, - the method will draw a lot of unwanted samples hence is not very efficient - becomes very problematic when the data is high-dimensional. - Even at dimensions of `\(\sim 10\)`, this method become very inefficient – i.e. you need to draw lots of samples before one sample is accepted --- class: segue-yellow background-image: url("https://raw.githubusercontent.com/garthtarr/sydney_xaringan/master/assets/USydLogo.svg") background-size: 20% background-position: 95% 95% # Monte Carlo simulation examples --- ## Collecting Coles or Woolworths kids toys * Collect one little shop when you spend $30 * How many $30 shops do I need to collect all 30 items? <img src="https://github.com/usydstat5003/stat5003-graphics/blob/master/Week10/coles.png?raw=true" width="25%"/> --- ## Theoretical aspect (before we see the Monte Carlo) Expected number of shops you need to do: `\begin{align*} \mathbb{E} T = n \left(\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}\right) \approx n \log n + 0.5772n + 0.5 \end{align*}` * where, `\(T\)` is the time to collect all items, `\(n\)` is the number of items to collect So, for a Coles little shop would expect, `\(\mathbb{E}(T) \approx 120\)` --- ## Monte Carlo simulation of Cole's little shop .columns-2[ .pull-left[ ```r n.shops.sim <- c() for(i in 1:1000) { collected <- c() n.shops <- 0 while(length(collected) < 30) { newitem <- sample(30,1) collected <- union(collected, newitem) n.shops <- n.shops + 1 } n.shops.sim <- c(n.shops.sim, n.shops) } mean.sim <- mean(n.shops.sim) ggplot(data.frame(x = n.shops.sim)) + theme_minimal() + geom_histogram(aes(x = x, y = ..density..), colour = "black", fill = "white") + labs(x = "Number of trips", y = "Density") + geom_vline(xintercept = mean.sim, linetype = "dotted", colour = "red") ``` ] .pull-right[ <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="90%" /> ] ] --- ## Polya urn problem * Let an urn start with 1 black ball and 1 white ball. * Repeatedly draw a single ball from the urn * Each time a ball is drawn - you put back that ball plus one extra ball of the same colour back into the urn. > After 1000 draws, how many balls of each colour do you expect to be in the urn? * This is an example of using Monte Carlo to simulate a process --- ## Monte Carlo simulation of the Polya urn problem <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Asymptotic Uniform distribution of the proportion of black balls <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> * In the limit the distribution is `\(U(0,1)\)` --- ## Simulate stock price as a random (stochastic) model Equation for the future price `\(S\)` of a stock: `\begin{align*} dS = S (\mu dT + \sigma \sqrt{dT} \mathcal N) \end{align*}` * `\(\mu\)` is the growth rate of the stock price * `\(\sigma\)` is known as the volatility. - Think of this as the variance of the stock price * `\(\mathcal N\)` is a normal random variable * `\(dT\)` is one unit of time --- ## Examples of a stock price <img src="data:image/svg+xml;base64,<?xml version='1.0' encoding='UTF-8' ?>
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" width="70%" /> --- ## Options A .brand-red[call] option gives you the right to buy a stock at a particular price at expiry > E.g. 90 day call option with $105 strike. Let’s say your stock is currently priced at $100, if it hits $108 at day 90, then you can exercise the option and make a $3 profit. If its price is below $105 at day 90, then you make nothing. A .brand-red[put] option gives you the right to sell a stock at a particular price at expiry --- ## Option pricing with Monte Carlo * How much would you pay for a call option? Think of this as the amount of insurance premium. * Using Monte Carlo simulation, simulate many realisations of the stock price. For each realisation, calculate the option payoff * The price you should pay for the option is the expected value of the profit you should make --- ## References Shonkwiler, R. W. and F. Mendivil (2009). _Explorations in Monte Carlo Methods_. Springer Science & Business Media.